Definition 1.1 A sequence of pointed smooth manifolds \((M_i, g_i, p_i) \)  with or without boundary is said to converge, up to diffeomorphisms, to \((M,g,p)\)  in the pointed flat metric topology, if we can find diffeomorphisms \(\varphi _i: M_i \rightarrow \varphi _i(M_i)\subset M\)  with \(\varphi _i(p_i)=p\)  such that \(\varphi _{i,\#}M_i \rightarrow M\)  in the pointed flat metric topology as currents on \(M\)  (see Definition 31.1 in [1]}). In other words, for any fixed \(r>0\)  and any coordinate domain \(x:U\rightarrow \mathbb {R}^n\)  with \(U \subset B(p,r) \subset M\) , we have
\(\lim _{i\rightarrow \infty }\int _{U}\left| \chi _{\varphi _i(M_i)}\sqrt{\det g_i} - \sqrt{\det g} \right| dx =0.\) 

